Incomplete quadratic exponential sums in several variables

We consider incomplete exponential sums in several variables of the form S(f,n,m)=12n∑x1∈{-1,1}⋯∑xn∈{-1,1}x1⋯xne2πif(x)/p,where m>1m>1 is odd and f is a polynomial of degree d   with coefficients in Z/mZZ/mZ. We investigate the conjecture, originating in a problem in computational complexity, that for each fixed d and m   the maximum norm of S(f,n,m)S(f,n,m) converges exponentially fast to 0 as n   tends to infinity; we also investigate the optimal bounds for these sums. Previous work has verified the conjecture when m=3m=3 and d=2d=2. In the present paper we develop three separate techniques for studying the problem in the case of quadratic f  , each of which establishes a different special case. We show that a bound of the required sort holds for almost all quadratic polynomials, the conjecture holds for all quadratic polynomials with n⩽10n⩽10 variables (and the conjectured bounds are sharp), and for arbitrarily many variables the conjecture is true for a class of quadratic polynomials having a special form.